How Machine Learning Studies Correlation?
Welcome to AI.SE @Laura!

It sounds like you have a series of data points, each with 10 related measurements, and you want automatically assess which of the measurements are most closely related to each other.

You are right that the correlation coefficient is a good choice for this.

Other techniques used in some AI algorithms include the Information Gain measurement (where you measure the reduction in entropy of one variable that follows from partitioning on another one), and embedded feature selection approaches, like the one in this paper.